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the circle is equal to

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the same with the velo

city at the mean distance in the ellipsis.

Cor. 5. If a body revolve in a circle, whose radius is equal to the aphelion or perihelion distance of a planet, the velocity of the planet revolving in the ellipsis at these points, will be to the velocity of a body revolving in the circle, as the square root of the latus rectum of the el lipsis, to the square root of the diameter of the circle. Because, in these points, the tangents of both curves, and the perpendiculars to the tangents coincide, and the latus rectum of the circle is equal to its diameter. Therefore the velocities being proportional to the square roots of the latera recta, when the perpendiculars are the same, they are to one another as the square root of the latus rectum of the ellipsis, to the square root of the diameter of the circle. V:v:: L: 2R.

PROPOSITION X.

If the central force, which causes an ellipsis to be described round its focus, should decrease more or less in proportion, than as the square of the distance of the revolving body increases, the body would leave the ellipsis, unless the curve itself be supposed to revolve on its focus; and on this hypothesis, the greater axis would move in the same direction with the body, when the central foree becomes too small, and in a contrary direction, when it becomes too great, to be in proportion to the squares of the distances inversely.

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The law of the central force residing in the focus of an ellipsis, being, to increase in strength, as the squares of the distances decrease; if any alteration should hap pen in this law, the distances of the revolving body will be either greater or less, than what they would be, with

out such an alteration, and consequently the revolving body will be either without or within a fixed and immovable ellipsis. Suppose that the body moves from the aphelion towards the perihelion, and if the central force do not increase so fast as the squares of the distances decrease, the central force will attract the body with less power, and therefore suffer it to move without the ellipsis, unless by revolving on its focus, the superior apsis of the ellipsis advance in the same direction with the body, and with a velocity proportioned to the want of due strength in the central body, so as that some point of the curve may still coincide with the place of the body.

And on the contrary, if the central force increased faster than the squares of the distances decreased, it would urge the revolving body too forcibly towards the focus, and cause it to move within the ellipsis; unless the superior apsis receded backwards, and thereby caused some point of the ellipsis nearer to the focus to coincide with the place of the body. So that in the first case, when the central force is too weak, the superior apsis moves forward in the same direction with the revolving body; and in the other case, when it is too strong, the superior apsis will have a retrograde motion in a contrary direction.

We shall have occasion for this proposition in 'astronomy, to explain the motion of the apsis of the moon. For although the central force of the earth, which retains the moon in her orbit, would make her describe an immovable ellipsis round the earth, placed in one of its foci, if she were not attracted by any other force; yet. we shall see, that she is also attracted by the sun, and this additional force, sometimes conspiring with, and sometimes acting against, the force of the earth, greatly

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disturbs her motion. At the new and full moon, the force, with which the sun acts upon it, causes its gravity towards the earth to decrease, in the departure of the moon, and to increase in its approach, faster than in the inverse proportion of the squares of the distances. From whence it will follow, that as the attraction of the sun is contrary to that of the earth, at the new and full moon, the attraction of the earth will be too weak to retain her in an immovable ellipsis; her distance from the earth will be increased, and therefore the apsis of her orbit will move forward, in the same direction with her own motion, which is from west to east.

But at the quarters, the disturbing force of the sun, conspiring with the attractive force of the earth, urges the moon more forcibly towards the focus, and causes her to descend within the ellipsis which she began to describe at her higher apsis; and therefore her orbit is to be considered, at these points, as a movable ellipsis, whose apogee recedes backwards in a direction contrary to her own motion.

HYDROSTATICS.

HYDROSTATICs is that branch of natural philosophy, which explains the nature, gravity, pressure and motion of fluids, whether elastic or nonelastic.

Although nothing seems, at first sight, more easily determined than the essential distinction between a fixed and fluid body, and every person sees the difference between them, yet the more that this subject is examined, the more difficulties occur in the determination. They both evidently consist of the same kind of matter, and the original constituent particles of both are the same, as they are so easily and frequently changed from one form to the other, as in the fusion of metals

and freezing of water. Hence, the ancients supposed, that the difference consisted in this; that a fixed body retained its figure, while the fluid, unless confined in some vessel, exerted a continual influence to change its form. This however cannot be the true definition of a fluid, because it will also agree with a fine powder, which none ever called a fluid.

Descartes thought, that he had given a much better definition of a fluid, when he supposed, that the essence of fluidity consisted in the intestine motion of the particles; however this motion might be imperceptible by

our senses.

Yet, although we grant, that the particles of a fluid have a constant tendency to motion, it is still a hasty conclusion, that they are actually in motion.

Sir Isaac Newton and his followers, rejecting the definition of Descartes, have endeavoured to ascertain the nature of a fluid from the principles of mechanics, and have determined that fluidity consists in these three qualities following.

1. The exceeding smallness of the particles: 2. Their roundness, or figure necessary for constituting volubility: 3. A very slight cohesion, or rather none at all. But these conditions are not of themselves sufficient to constitute fluidity; for, unless we add the gravity of each particular particle, there cannot be a tendency to flow. Yet all these four conditions are found in the calx of tin and minium, when sufficiently pulverized; and still they do not flow, until they begin to vitrify with a sufficient degree of heat.

To avoid this difficulty, our modern philosophers, neglecting every other property of a fluid, excepting its cohesion and gravity, say, that a fluid is that, whose smallest particles do not cohere with a greater force,

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than what is equal to the weight of a single drop. But still it may be inquired, what are these smallest parti. cles, which cannot be divided into lesser?

As matter is in its own nature infinitely divisible, it is impossible to assign its smallest particles. Besides, it is evident to our senses, that the force of cohesion in small drops is much greater than their gravity, and that they do not separate until, by the constant accession of other particles, they increase to such a magnitude, that the gravity of the whole drop is more than sufficient to be a counterbalance to their cohesion.

These things then, failing to give satisfaction to the philosophic mind, let us attend to the operation of nature, and see, if we cannot investigate the precise dif ference between a fixed and a fluid body, from some of the most obvious properties of a fluid.

1. At first sight, there is this remarkable difference between a fixed and fluid body, that the particles of the former are so closely connected together, that they exert only one common force by their gravity; but besides this force of gravity, which all fluids possess in common with fixed bodies, the particles of a fluid are not so strictly bound together, but that they are at liberty to exert a very remarkable force upon each other. And hence it is that the particles of a fluid constantly tend to an equilibrium, and, when unrestrained by an external force, compose themselves into a plane surface; which we never find to be the case with powders, however levigated and subtile their particles may be.

2. Although the particles of a fluid are not so strictly connected together as to hinder a remarkable exertion of force upon each other, yet they have a considerable cohesion and tendency to union, when the fluids are unelastic; and this cohesion is greater in some than in

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